Dynamic Stability Analysis of Blunt-Body Entry Vehicles Using Time-Lagged Aftbody Pitching Moments

Abstract

This analysis defines an analytic model for the pitching motion of blunt bodies during atmospheric entry. The proposed model is independent of the pitch-damping sum coefficient present in the standard formulation of the equations of motion describing pitch oscillations of a decelerating blunt body, instead using the principle of a time-lagged aftbody moment as the forcing function for oscillation divergence. It is shown that the dynamic oscillation responses typical to blunt bodies can be produced using hysteresis of the aftbody moment in place of the pitch-damping coefficient. Four parameters, all with intuitive physical relevance, are introduced to fully define the aftbody moment and the associated time delay. The approach used in this investigation is shown to be useful in understanding the governing physical mechanisms for blunt-body dynamic stability and in guiding vehicle and mission design requirements. A validation case study using simulated ballistic range test data is conducted. From this, parameter identification is carried out through the use of a least-squares optimizing routine. Results show good agreement with the limited existing literature for the parameters identified, suggesting that the model proposed could be validated by a limited experimental ballistic range test series or with existing data. The trajectories produced by the identified parameters are found to match closely those from the Mars Exploration Rover ballistic range tests for a range of initial conditions. Nomenclature A = Euler-Cauchy angle-of-attack coefficientaerodynamic pitching moment slope coefficient d = aerodynamic reference diameter, m g = acceleration due to gravity, m∕s 2 h = altitude, m I yy = pitch axis mass moment of inertia, kg · m 2 l = characteristic length, m M = Mach number m = mass, kg N = number of simulated ballistic range shots R p = planet radius, m S = cross-sectional area, m 2 t = time, s t lag = lag time, s t − = time referenced by aftbody, s V = vehicle velocity, m∕s v = characteristic velocity, m∕s x c:g: = axial location of center of gravity α = angle of attack, rad β = parameter of aftbody moment Mach number dependence γ = flight-path angle, rad δ = phase-shift constant, rad ε = residual θ = pitch angle, rad μ = Euler-Cauchy oscillation growth exponent ν = Euler-Cauchy frequency coefficient ρ = atmospheric density, kg∕m 3 τ = lag time factor Subscripts eq = equivalent 0 = initial quantity ∞ = freestream Superscripts AB = aftbody contribution FB = forebody contribution * = reference value for aftbody moment curve

Dynamic CFD Simulations of the MEADS II Ballistic Range Test Model

“…This "phase lag" phenomenon has been cited as one of the mechanisms of dynamic instability in entry capsules. For a more detailed treatment of this theory, the reader is referred to the work of Abe et al, 3 Teramoto et al, 4 and Kazemba et al 5 For the current analysis it is sufficient to say that, based on this theory, the greater the phase offset, and the greater the amplitude of the backshell moment, the greater will be the instability. Comparing the two plots, we do not see significant difference in the extent of phase lag between the two codes.…”

“…1,2 The current ballistic range campaign will provide unique data for comparison to CFD predictions in that the ballistic range models will be instrumented with pressure transducers, enabling greater resolution of the forebody and aftbody contributions to the total aerodynamic forces. Additionally, pressure transducers located on the model backshell will allow comparison of the predicted wake flow environment, which is believed to have a significant effect on the dynamic stability of blunt bodies in the low-supersonic to transonic regime, [3][4][5] to that of the experiment.…”

Dynamic instability modeling on phase plane for thin aeroshell capsule with pitching motion at transonic speed

“…In the free-flight test results of the Hayabusa-type and wind tunnel test results, despite variations in some conditions, reports indicated that the frequency changed, while the AOA amplitude remained relatively consistent. Abe et al (2000) and Kazemba et al (2015) extended the Hiraki model to account for the phase delay, thereby allowing for an assessment of how the phase delay influenced the parameters of the model. Additionally, coupled analyses (Takeda et al, 2020;Yang et al, 2016;Stern et al, 2012) with fluid and motion were used to simulate wind tunnel test results and improve understanding of dynamic instability.…”

Coupled Numerical Analysis of Three-Dimensional Unsteady Flow with Pitching Motion of Reentry Capsule—Investigation of the Third Harmonics of the Aerodynamic Force—

“…Hiraki et al 2,3) developed a model equation for the aerodynamic force based on the van der Pol equation. Abe et al 4) and Kazemba et al 1,5) established model equations introducing phase delay effects called "temporal delay" and "lag time factor," respectively, with that of Abe et al being an extended Hiraki model. Although these model equations reproduced well most of the pitch-angle oscillations in the experiments, the third harmonics of the pitching moment was not sufficiently discussed.…”